If two ideas, for example, are similar, the implication is there really are two ideas - that have many points in common (though sometimes the points in common may be very context-specific).
If two ideas are the same, the implication is there's actually only one idea (albeit perhaps presented in two very different forms).
This distinction is commonly understood. Thus, for example...
1: John's and Peter's essays both make the same point.
2: John's and Peter's essays both make the same points.
...are both valid. In (1), the implication is one single point is made by both essays (either or both essays may make additional points which aren't in the other). In (2), the implication is more than one point is made in both essays.
Thus, there's nothing wrong with saying "We both have the same ideas" - it just means we have more than one idea in common. But no-one ever says ?"We both have two same ideas". If the meaning is there's one idea [which we've each thought of independently], it's "We both have the same idea".
If the intended meaning is we've each independently thought of "Idea A" and "Idea B", English doesn't really have a succinct unambiguous way of expressing the concept using the word same. You'd probably rephrase to something like "We have two ideas in common", or "We share two ideas".
That's at the semantic level. At the grammatical level, all I can say is standard usage is at least consistent - we never say ?"This is a same idea", or ?"Here are three same coins". If several things are the same, there's really only one of "it" (within the current "frame of reference"). That's why it's singular, and why it can't take the indefinite article (to speak of "an X" implies the existence of multiple X's).